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On normal structure, fixed-point property and contractions of type $ (\gamma)$

Author: M. A. Khamsi
Journal: Proc. Amer. Math. Soc. 106 (1989), 995-1001
MSC: Primary 46B20; Secondary 47H10
MathSciNet review: 960647
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Abstract: We prove that a Banach space $ X$ has normal structure provided it contains a finite codimensional subspace $ Y$ such that all spreading models for $ Y$ have normal structure. We show that a Banach space $ X$ is strictly convex if the set of fixed points of any nonexpansive map defined in any convex subset $ C \subset X$ is convex and give a sufficient condition for uniform convexity of a space in terms of nonexpansive map of type $ \left( \gamma \right)$.

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  • [1] B. Baillon, Comportement asymptotique des contractions et semi-groupes de contractions--Equation de Schrödinger nonlinéaires et divers, Thèses présentées à l'université Paris VI, 1978.
  • [2] B. Beauzamy et J. T. Lapreste, Modèles étalés des espaces de Banach, Publication du Département de Mathématiques de l'Université Claude Bernard-Lyon I 1983.
  • [3] M. S. Brodskiĭ and D. P. Mil′man, On the center of a convex set, Doklady Akad. Nauk SSSR (N.S.) 59 (1948), 837–840 (Russian). MR 0024073
  • [4] Ronald E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math. 32 (1979), no. 2-3, 107–116. MR 531254,
  • [5] Mahlon M. Day, Normed linear spaces, 3rd ed., Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21. MR 0344849
  • [6] Ralph DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math. 13 (1963), 1139–1141. MR 0159229
  • [7] J. R. Giles, Brailey Sims, and S. Swaminathan, A geometrically aberrant Banach space with normal structure, Bull. Austral. Math. Soc. 31 (1985), no. 1, 75–81. MR 772632,
  • [8] M. A. Khamsi, Étude de la propriéte du point fixe dans les espaces de Banach et les espaces metriques, Thèses présentées à l'Université Paris VI, 1987.
  • [9] W. A. Kirk, Nonexpansive mappings and normal structure in Banach spaces, Proceedings of research workshop on Banach space theory (Iowa City, Iowa, 1981) Univ. Iowa, Iowa City, IA, 1982, pp. 113–127. MR 724109
  • [10] Thomas Landes, Permanence properties of normal structure, Pacific J. Math. 110 (1984), no. 1, 125–143. MR 722744
  • [11] Brailey Sims, “Ultra”-techniques in Banach space theory, Queen’s Papers in Pure and Applied Mathematics, vol. 60, Queen’s University, Kingston, ON, 1982. MR 778727
  • [12] S. Swaminathan, Normal structure in Banach spaces and its generalisations, Fixed points and nonexpansive mappings (Cincinnati, Ohio, 1982) Contemp. Math., vol. 18, Amer. Math. Soc., Providence, RI, 1983, pp. 201–215. MR 728601,
  • [13] V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. Rozprawy Mat. 87 (1971), 33 pp. (errata insert). MR 0300060

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Keywords: Nonexpansive mappings, normal structure, fixed point property
Article copyright: © Copyright 1989 American Mathematical Society

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